A Psychologically Suitable Measure Of Investment Performance

Measuring the financial performance of an asset, portfolio or strategy is necessary for comparing investment choices or optimizing a trading algorithm. Because no sound investment decision can focus on the returns while forgetting the associated risks entirely, academics have proposed to combine both in so-called 'risk-adjusted returns'. Since the last four decades, a plethora of ratios have been invented, some becoming ubiquitous despite their now apparent weaknesses, some others remaining unfairly obscure. Yet, for all their mathematical prowess, we think none of these ratios constitutes a satisfying performance indicator from the individual investors' perspective:

Relative performance is psychologically more relevant than absolute performance. It is our strong belief that investors at large psychologically anchor their decisions on the overall stock market.

While not reflected by statistical measures such as mean or volatility, the sequencing of returns are determinant for investors. Imagine a strategy loosing 10% per year for a decade, then suddenly earning 500%. Most people would loose faith and abandon it before it becomes fruitful.

Investors are loss-averse. Behavioral experiments have indicated people tend to value two or three times more what they loose than what they gain (see Gächter, Johnson & Hermann, 2010)

We'll start by comparing existing methods for measuring those two components of Investment performance: the Return, and the Risk.

Calculating the return

The return looks straightforward, and can be described as the amount by which an investment grew and generated cash-flow. It is also called 'Rate of Return', 'Return on Investment', 'yield' or 'arithmetic return'. Without taking external cash-flow into account, this value is calculated as (final value / initial value) - 1. Returns are not duration-independent (earning 10% in one month is quite different from earning 10% over 5 years), and have to be 'normalized' to be comparable. Usually they're normalized over a one-year period, hence the term 'annualize'. Because of the law of compounding returns, if r is the rate of return and N/n is the ratio between the number of periods in one year and the number of observed periods, then R the annualized return equals (1+r)^(N/n) - 1. For instance, 15% in 18 months equals: (1.15)^(12/18)-1 = 9.77% annualized. It is generally a bad idea to extrapolate short-term results by annualizing them. An annualized 7% return over one week would give an impressive but totally unrealistic return of 3,273% per year...

Another approach is to measure the Excess Return, which is the return relative to a given benchmark. It is calculated by subtracting the benchmark return from the investment return. A common benchmark reference is a virtually risk-free investment, such as Treasury Bills.

Measuring the risk

The notion of risk is fuzzier. For most quantitative traders, risk equals volatility, which in plain terms means 'how much do the returns vary'. The common idea that any volatility is bad, since investors don't like uncertainty.

The Sharpe ratio is the best-know measure of risk-adjusted performance relying on this idea. It is defined as the average of the excess returns divided by the volatility, or standard deviation of those excess returns. The Sharpe ratio consider a normal distribution of returns (plotting the returns draws a nice bell curve). However, real-life returns show much fatter tails, invalidating that assumption.

Intuitively, another problem of the Sharpe ratio lies in its definition of risk. It's hard to find an investor unhappy about suddenly making more money than he expected, yet the Sharpe ratio puts an equal penalty on positive or negative deviations.

The Sortino ratio tries to address that by taking only the downward volatility into account. Unfortunately, it makes the Sortino Ratio more sensitive to noise, since it takes less data points into account.

Another approach of risk is to measure the drawdowns. A drawdown is the percentage below a maximum value previously achieved. In other words, a drawdown occurs every time a price goes down, and lasts until the price went back to its previous highest level. the Maximum Drawdown is the largest of those drawdowns, or losses in value. Popular ratios taking drawdowns into account are the Sterling Ratio and the Calmar Ratio. Roughly speaking, they divide the return by the Maximum Drawdown. Unfortunately these ratios don't factor in the recovery time. Another glaring omission is that they take only the biggest drawdown into account. In reality, the number and duration of drawdowns are psychologically almost as important as their maximum amplitude. Furthermore, because the Sterling and Calmar ratios are based on a single, worst event, they are not very reliable from a statistical point of view. Martin & McCann proposed a clever alternative in 1989, that they cunningly named the 'Ulcer Index', and which is the quadratic mean of the drawdowns. In other words, it's the square root of the arithmetic average of the square of the drawdowns. Squaring the drawdowns is in direct inspiration of the standard deviation calculus of the Sharpe ratio, and is designed to over-penalize large losses compared to small ones. A 20% loss is more worrying than two 10% losses. The derived risk-adjusted performance measurement is called the Martin Ratio, and is the excess return divided by the Ulcer Index.

Yet another approach is to sum the negative returns. Although the Sortino ratio looks like an adaptation of the Sharpe ratio, it belongs to another class of measures, called Kappa ratios by Kaplan & Knowles. The best known member of the Kappa ratios is called the Omega ratio, and is generally viewed as the probability of a gain divided by the probability of a loss. The Omega ratio can be calculated as the ratio between the sum of the returns above a minimum acceptable level and the sum of the returns below that level.

The V2 ratio

While the Martin and Omega ratios are effective and straightforward risk-adjusted performance measurements, they still don't address the problem of relative performances. Most investors being down by 3% while the market lost 12% would feel relieved, almost satisfied. Likewise, an investor making 15% while the market is up by a staggering 44% will feel like he's losing. Comparing the Martin or Omega ratios of two different assets won't solve the problem, since it doesn't tell us if the ups and downs occur at the same time.

Therefore we propose a new kind of measurement, the V2 ratio. 'V2' stands for 'Valu Valu', our company name, as we originally designed this ratio to optimize our quantitative trading strategy. The V2 ratio is an attempt to combine the relativity of the Sharpe Ratio with the consistency of the Omega Ratio. We also wanted the V2 Ratio to have the following properties:

The V2 Ratio of a investment that systematically beats the benchmark should be greater or equal to the excess return.

An investment with a return inferior to the Benchmark should have a negative V2 Ratio.

The V2 Ratio should always be a valid number, unlike the Martin or Omega ratios that can't be computed if there's no drawdowns or underperformance.

The V2 Ratio of the Benchmark should be zero.

The V2 Ratio should over-penalize big underperformances

Calculation

The V2 ratio is calculated as the ratio between the annualized excess return and the quadratic mean of the relative drawdowns plus one. The relative drawdowns are equal to the drawdowns of the investment minus the drawdowns of the benchmark. For instance, if a stock price goes from $45 to $41 while the S&P500 goes from $1260 to $1230, the relative drawdown is (41/45 - 1) - (1230/1260 -1) = 41/45 - 1230/1260 = -8.9% + 2.4% = -6.5%. The stock's bad performance is mitigated by the poor performance of the market in general.

The V2 ratio formula is:

Where R is the investment return, Rb the benchmark return, pi the investment price at time i, Pi the highest investment price reached at time i, bi the benchmark price at time i, Bi the highest benchmark price reached at time i and n the number of periods.

At last, here are a few examples of ratio calculations, based on the weekly prices of stocks between January 2004 and October 2011 and taking the S&P500 as benchmark. It is worth going through each stock and seeing how the V2 ratio compares to better known indicators:

If human beings were perfectly rational, the only thing that would matter would be total returns. But they get scared, impatient or just worried. The V2 Ratio is not perfect: it's new, slightly more complicated to compute than better-known ratios. But it is a necessary attempt to provide a measure more in sync with investors' feelings. Behavioral economics have moved from the Utility theory to the Prospect theory, maybe it's time for performance indicators to do the same.

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.